(x-2y).jpeg "(x^2-2xy+y^2)(x-2y)")
Expanding the Expression (x² - 2xy + y²)(x - 2y)
This article explores the process of expanding the given expression: (x² - 2xy + y²)(x - 2y).
Recognizing Patterns
Before diving into the expansion, let's observe the given expression. The first factor (x² - 2xy + y²) is a perfect square trinomial, specifically (x - y)². The second factor (x - 2y) is a simple binomial.
Expanding using the Distributive Property
We can expand the expression using the distributive property:
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Multiply the first term of the first factor (x²) by each term in the second factor:
- x² * x = x³
- x² * (-2y) = -2x²y
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Multiply the second term of the first factor (-2xy) by each term in the second factor:
- -2xy * x = -2x²y
- -2xy * (-2y) = 4xy²
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Multiply the third term of the first factor (y²) by each term in the second factor:
- y² * x = xy²
- y² * (-2y) = -2y³
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Combine all the terms: x³ - 2x²y - 2x²y + 4xy² + xy² - 2y³
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Simplify by combining like terms: x³ - 4x²y + 5xy² - 2y³
Conclusion
Therefore, the expanded form of (x² - 2xy + y²)(x - 2y) is x³ - 4x²y + 5xy² - 2y³. This process showcases the application of the distributive property and recognizing patterns to simplify algebraic expressions.